Overview

Unlike many libraries, S2 fully supports degenerate geometry. Degeneracies not only do not cause errors, they have well-defined semantics including support for open, semi-open, and closed boundaries. Supported degeneracy types include the following:

• Point polylines consisting of a single degenerate edge AA (where A represents a vertex).

• Point loops consisting of a single vertex A. Such loops may represent either point shells or point holes according to whether the loop adds to or subtracts from the surrounding region of the polygon.

• Sibling edge pairs of the form {AB, BA}. Such sibling pairs may represent either degenerate shells or degenerate holes according to whether they add to or subtract from the surrounding region. The edges of a sibling pair may belong to the same polygon loop (e.g. a loop AB) or to different polygon loops (e.g. the polygon consisting of the loops ABC and CBD).

Supporting such degeneracies has one big advantage, namely that geometry can be simplified without completely losing small details. For example consider a polygon representing a land region with many small lakes. Each lake would be represented as a hole. When this polygon is simplified, some or even all of the lakes may collapse to single points. Without support for degeneracies, such lakes would disappear from the simplified result. If we allow degeneracies, on the other hand, we can guarantee that for every point in the original region there is a nearby corresponding point in the simplified region and vice versa. In other words every lake in the input is still near some lake in the output.

Dimension Reduction: A Poor Substitute

An alternative technique for handling degeneracies is dimension reduction, whereby degenerate geometry of dimension 2 or 1 is replaced by non-degenerate geometry of dimension 1 or 0. For example, the sibling pair {AB, BA} would be replaced by a polyline edge (either AB or BA, chosen arbitrarily), and similarly the degenerate polyline AA would be replaced by a single point A.

This technique has four very significant drawbacks compared to true degeneracy support as implemented by the S2 library:

• Dimension reduction is only capable of handling positive degeneracies (e.g., polygon shells) as opposed to negative degeneracies (e.g., polygon holes). Consider again the example of simplifying a land region with many small lakes that collapse to single points. Since these lakes are represented as holes in the polygon rather than shells, they cannot be replaced by points but instead must be simply discarded. Similarly, dimension reduction is not capable of representing negative degeneracies of dimension 1 (e.g., a land polygon containing a river that when simplified has its width collapse to zero).

• Dimension reduction changes the boundary of the affected geometry. In general the boundary of a polygon is defined to consist of its edges, the boundary of a polyline is defined to consist of its two endpoints, and points are defined to have no boundary at all. This implies that when the degenerate sibling pair {AB, BA} is replaced by the polyline edge AB, for example, the definition of its boundary changes from the two edges {AB, BA} to the two points {A, B}. This is a very significant difference, since it can change the result of virtually all the spatial relationship predicates defined by OGC Simple Features model (which are based on evaluating whether the interior, boundary, and exterior of one region intersect the interior, boundary, and exterior of the other as summarized by the DE-9IM matrix).

• Dimension reduction forces clients to deal with heterogeneous geometry as a possible result of virtually any geometric operation. For example, intersecting two polygons may yield a collection of geometry consisting of polygons, polylines, and points. Similarly, intersecting two polylines may yield a collection of polylines and points. In contrast, the true degeneracy support offered by S2 means that any boolean operation on two polygons is guaranteed to yield a polygon, and any boolean operation on two polylines is guaranteed to yield a polyline. This is not only simpler for clients but also reduces the opportunities for bugs (since the cases resulting in mixed-dimension geometry may be unexpected and rare).

• Dimension reduction makes it hard for clients to say how degeneracies should be handled. Some clients may wish to discard degeneracies, generate errors, or convert degeneracies back into non-degenerate forms (e.g. replacing point shells with tiny triangles). True degeneracy support makes it easy to implement any of these models, and even to delegate this decision to the functions that convert geometry to non-degeneracy-supporting formats (such as geoJSON). Dimension reduction, on the other hand, makes it difficult to distinguish degeneracies from legitimate lower-dimensional objects. For example, imagine intersecting two collections of polylines and polygons. Some output polylines will correspond to portions of input polylines, while others may correspond to intersecting polygons that abut along an edge.

Other alternatives, such as extracting small details into separate objects to protect them from simplification and then merging them back in afterwards, can best be described as hacks. Not only are such algorithms virtually impossible to make robust, they cannot offer the mathematical guarantees that can be achieved by simplifying with a uniform error tolerance using degeneracies (see below).

Degeneracies and Hausdorff Distance

The fact that S2 supports both positive and negative degeneracies allows us to make a very powerful guarantee when simplifying geometry. Recall that the Hausdorff distance between two geometries A and B is defined as

$H(A, B) = {\rm max} \{ h(A, B), h(B, A) \}$

where $$h(A, B)$$ is the directed Hausdorff distance

$h(A, B) = {\rm max}_{a \in A} \big\{ {\rm min}_{b \in B} \{ d(a, b) \} \big\} .$

The Hausdorff distance is a measure of how different two geometries are; in particular it is an upper bound on how far away a point in one geometry can be from the other geometry.

The most important property of Hausdorff distance is that it can be used to bound the distance measurement errors due to simplifying geometry. Suppose that geometry B is a simplified version of geometry A, and that we wish to measure the distance to some third geometry X. Then the error due to simplification, i.e. the difference between $$d(A, X)$$ and $$d(B, X)$$, is bounded by the Hausdorff distance between A and B:

$| d(A, X) - d(B, X) | \leq H(A, B) \, .$

We can now state the guarantee that the S2 library makes when simplifying geometry. Let geometry A be an arbitrary collection of points, polylines, and polygons, and suppose that it is simplified using S2Builder with a snap radius of r to yield a new geometry collection B. Then if degeneracies are allowed when constructing B, we can guarantee that

$H(A, B) \leq r_{\rm e} \, .$

The quantity $$r_{\rm e}$$ is called the maximum edge deviation, which S2Builder ensures is at most 10% greater than the given snap radius r (i.e., $$r_{\rm e} \leq 1.1 r$$). In other words, the maximum possible distance measurement error due to simplification is only slightly greater than the snap radius used during simplification. (Note that the additional 10% is a guaranteed bound rather than a heuristic; it is only necessary because S2 works with spherical geometry and would not be needed in a planar version of the algorithm.)

In fact, under the view that positive and negative degeneracies have infinitesimal areas, the bound above applies not only to the whole geometries A and B, but also to the interiors, boundaries, and exteriors of these geometries:

$H({\rm Int}\, A, {\rm Int}\, B) \leq r_{\rm e}$ $H(\partial A, \partial B) \leq r_{\rm e}$ $H({\rm Ext}\, A, {\rm Ext}\, B) \leq r_{\rm e} \, .$

These properties are the holy grail of simplification algorithms; it is a mathematical way of saying that no details larger than the maximum edge devation $$r_{\rm e}$$ have been lost.

Representation and Validity

Degenerate geometry is not supported by the legacy classes S2Polyline and S2Polygon. Instead the more modern, efficient, lightweight classes S2LaxPolylineShape and S2LaxPolygonShape must be used. (The Lax in the names refers to the fact that these classes allow degeneracies.) For example, while S2Polygon requires that all loops have at least three vertices, S2LaxPolygonShape supports loops with two, one, or even zero vertices. (The zero-vertex case is called the full loop and represents the full sphere.)

The Lax classes themselves do not have any validity requirements, but the geometry they represent must satisfy certain conditions before it can be used in S2 operations. Some operations are more restrictive than others; for example, distance measurement has fewer requirements than S2BooleanOperation. You will generally want to construct your geometry to satisfy the most restrictive rules, outlined below. This can easily be accomplished by using S2Builder with an appropriate output layer type (e.g. S2LaxPolygonShape).

Polylines

Polylines must consist of either a single degenerate edge AA, or a sequence of non-degenerate edges (e.g., ABC). Polylines with both degenerate and non-degenerate edges are not allowed (e.g., AABC or ABBC). On the other hand, polylines may self-intersect or have duplicate edges.

Note that a point polyline is represented as two vertices (AA) whereas a point polygon loop is represented as one vertex (A). Each of these objects thus consists of a single degenerate edge AA, corresponding to the general rule that an n-vertex loop has n edges, whereas an n-vertex polyline has n-1 edges.

Here are a few examples of valid and invalid polylines:

• AB: a valid polyline defining a single directed edge.

• ABCD: a valid polyline consisting of three directed edges {AB, BC, CD}

• [] (the empty vertex sequence): a valid polyline defining no edges.

• AA: a valid polyline defining a single degenerate edge at point A.

• ABBC: invalid because degenerate and non-degenerate edges are used in the same polyline.

• A: invalid because it defines a vertex but no edges. (A degenerate edge at A is represented as AA, while an empty polyline is represented as [].)

Polygons

Polygons must consist of a set of oriented loops such that the interior of the polygon is always on the left. Polygon edges may intersect only at their endpoint vertices. A polygon may contain both an edge and its reverse edge (i.e., AB and BA), but duplicate edges are not allowed (AB and AB). Loops also must not contain repeated adjacent vertices (e.g., ABBC).

Point loops consisting of a single vertex (A) are allowed, but such loops cannot be incident to any other edges. The full loop (i.e., the loop with zero vertices mentioned above) is allowed only if all the remaining loops (taken together) define only degeneracies.

Here are a few examples of valid and invalid polygons:

• {} (the empty set of loops): a valid polygon containing no points (the empty polygon).

• {A, BC}: a valid polygon consisting of one point shell and one sibling pair shell.

• {AB, BAC}: invalid because the edge BA occurs twice (note that loop AB consists of the two edges AB and BA).

• {ABC, A}: invalid because the point shell A is incident to other edges.

• {full}: a valid polygon containing the entire sphere (the full polygon).

• {full, ABCB}: a valid polygon consisting of the entire sphere except for two sibling pair holes (AB and BC).

• {full, ABC}: invalid because the full loop is used together with non-degenerate geometry.

• {full, ABC, ACB}: a valid polygon consisting of the entire sphere except for three sibling pair holes (AB, BC, and CA}. Note that even though the loops ABC and ACB are themselves non-degenerate, together they define only degenerate geometry and therefore can be used with the full loop.

Mixed-Dimensional Geometry

The S2ShapeIndex class represents an arbitrary collection of points, polylines, and polygons. Most S2 operations (e.g. S2ClosestEdgeQuery, S2RegionCoverer, S2ContainsPointQuery) do not impose any additional validity requirements on S2ShapeIndex beyond requiring that its component shapes are valid. The notable exception to this rule is S2BooleanOperation, which requires that polygon interiors must be disjoint from all other geometry (including other polygon interiors). So for example, polygon interiors must not intersect any point or polyline geometry.

In order to support degeneracies, S2BooleanOperation requires in addition that duplicate polygon edges are not allowed. Note that this rule is only necessary when an S2ShapeIndex contains more than one polygon, since duplicate edges are already disallowed within individual polygons. This additional rule essentially requires that the collection of all polygons in each S2ShapeIndex must satisfy the same validity requirements as a single polygon. (Also note that more than one polygon is never needed, because polygons in S2 can have more than one outer shell and therefore can represent an arbitrary subset of the sphere.)

Note that although duplicate polygon edges are not allowed, duplicate points and polyline edges are permitted. So for example, an S2ShapeIndex containing the points {A, A}, the polyline ABCABC, and the degenerate loop AB is a valid input to S2BooleanOperation. This is true whether the boundary model is open, semi-open, or closed. However an S2ShapeIndex input containing two polygons each consisting of the degenerate loop AB would be invalid.

Supporting points and polyline edges as multisets makes it much easier to support time series such as GPS tracks. For example, a car driving around a track might easily generate a polyline such as ABCABCABC. Similarly, this makes it easier to reconstruct geometry that has been snapped and/or simplified, which can cause distinct vertices to merge together. Note that if duplicate values are not desired, they can easily be removed by choosing the appropriate S2Builder output layer options.

Semantics of Degeneracies

In general degeneracies are modeled as infinitesimal regions of the appropriate dimension. So for example, a point shell A should be thought of as a tiny closed loop in the vicinity of the vertex A, and a degenerate shell loop AB (consisting of the two edges {AB, BA}) should be thought of as a narrow sliver in the vicinity of edge AB. Similarly, the polyline AA should be thought of as a tiny polyline in the vicinity of vertex A. We say “in the vicinity of” because under certain boundary models, degeneracies may not contain their nominal vertices and edges. For example, under the open boundary model the point shell A does not contain the point A. Instead it should be thought of as a tiny loop near the point A, so tiny that it does not contain any representable points along its edges or in its interior.

This model implies that lower-dimensional degeneracies are infinitesimal subsets of higher-dimensional degeneracies. For example, the point A is vanishingly small compared to the degenerate polyline AA. This is because the point A is truly zero-dimensional, whereas the polyline AA represents a very small one-dimensional set. As another example, consider a point A, a closed polyline AA, and a closed point shell A. The intersection of the point A and the closed polyline AA consists only of the point A, because it is an infinitesimal subset of the polyline, and similarly the closed polyline AA intersected with the closed polygon A yields only the polyline AA.

It is also worth recalling that in the S2 library, polyline and polygon edges do not contain any interior points. For example, the closed polygon shell loop AB (consisting of the sibling edge pair {AB, BA}) does not contain any points except A and B. This is because the S2 library models all geometry as being infinitesimally perturbed such that edges do not pass through any representable points on the sphere. This technique is known as simulation of simplicity (Edelsbrunner and Muecke, 1990) and greatly reduces the number of special cases that need to be handled when implementing robust geometric algorithms.

Boundary Models

The S2 library supports modeling polylines and polygons as having open, semi-open, or closed boundaries. Here we briefly review these models and discuss the treatment of degeneracies within them.

Note that the polyline and polygon boundary models are specified independently. The PolylineModel class specifies whether polylines contain their start and/or end vertex, whereas the PolygonModel class specifies whether polygons contain their vertices and edges.

Also recall that the boundary model is specified for each operation rather than being a property of the geometry itself. This allows more flexibility; for example, if two geometries A and B intersect under the CLOSED boundary model but not under the OPEN boundary model, this would mean that the boundaries of the two geometries intersect but not their interiors.

Closed Boundary Model

A closed polyline contains all of its vertices. For example, the closed polyline ABC intersected with the three points {A, B, C} yields {A, B, C}.

Similarly, a closed polygon contains all of its vertices, edges, and reversed edges. So for example, the closed polygon ABC intersected with the points {A, B, C} yields {A, B, C}, and the closed polygon ABC intersected with the six polylines {AB, BA, BC, CB, AC, CA} yields {AB, BA, BC, CB, AC, CA}.

These rules also apply to degeneracies. So for example, a closed point polyline AA contains its vertex A, and a closed sibling pair shell AB contains its vertices {A, B} and its edges {AB, BA}.

Note that certain degeneracies have no effect on point or edge containment; we call such degeneracies redundant. In the case of the closed boundary model, degenerate holes are redundant because they do not exclude any points or edges. For example, consider a polygon ABC containing a point hole D. Since the boundary is closed, the polygon contains the point D whether or not the hole is present. (The degenerate hole is still considered to exclude an infinitesimal region from the polygon, it’s simply that this region does not contain any representable points.)

Redundant degeneracies are considered to be part of the polygon’s boundary and therefore still affect distance measurement to the boundary. In fact, they may be viewed as a method of expanding the polygon’s boundary without affecting its interior. They are treated in exactly the same way as non-degenerate regions in boolean operations; for example, a redundant degeneracy intersected or unioned with itself yields the original degeneracy, whereas a redundant degeneracy subtracted from itself yields the empty set.

Open Boundary Model

An open polyline contain all of its vertices except the first and last. For example, the open polyline ABC intersected with the three points {A, B, C} yields {B}. The only exception is if the polyline_loops_have_boundaries() option is false and the first and last vertices of a polyline are the same (e.g. ABCDA); in this case the first/last vertex of the polyline loop is defined to be contained and its boundary is defined to be empty. So for example, with this option the open polyline ABCA intersected with {A, B, C} yields {A, B, C}.

An open polygon contains none of its vertices, edges, or reversed edges. For example, the open polygon ABC intersected with the points {A, B, C} yields the empty set, and similarly the open polygon ABC intersected with the six polylines {AB, BA, BC, CB, AC, CA} yields the empty set.

These rules also apply to degeneracies. So for example, an open point polyline AA contains no points at all, and an open sibling pair shell AB contains no points or edges. In other words, just as degenerate polygon holes are redundant in the closed boundary model, degenerate polylines and degenerate polygon shells are redundant in the open boundary model. These types of degeneracies do not affect point or edge containment, but simply add to the boundary of the affected geometry. The geometry contains exactly the same set of points and edges whether these degeneracies are present or not, however since the degeneracies change the boundary they can still affect distance measurement.

Also note that geometric objects have exactly the same boundary under all three boundary models. So for example, the boundary of an open degenerate polyline AA is the point A, even though the polyline is defined not to contain that point.

Semi-Open Boundary Model

The purpose of the semi-open boundary model is to allow geometry to cover a region without gaps or overlaps.

In order to achieve this, a semi-open polyline contains all of its vertices except the last. For example, the semi-open polyline ABC intersected with the points {A, B, C} yields {A, B}. The major advantage of this model is that polylines can be split into pieces without affecting vertex containment. For example, the single polyline ABCDEFG contains exactly the same set of vertices as the three polylines {ABC, CDE, EFG}, and furthermore each vertex is contained by exactly one of these polylines.

Similarly, semi-open polygon point containment is defined such that if several semi-open polygons tile the region around a vertex, then exactly one of those polygons contains that vertex. This implies that a triangle ABC may or may not contain each of its vertices {A, B, C}. However it ensures the important property that if a set of polygons tiles the entire S2 sphere, then every point on the sphere is contained by exactly one polygon. This property can be very useful for certain algorithms.

Semi-open polygons contain all of their edges, but none of their reversed edges. For example, consider two abutting triangles ABC and CBD. The edge BC is contained by ABC but not CBD, and the edge CB is contained by CBD but not ABC. This property ensures when a set of polygons tiles the sphere, each polygon edge is contained by exactly one polygon.

This rule also ensures that if a polyline is intersected with a set of polygons that tile the sphere, the resulting polylines can be unioned to obtain the original polyline without gaps or overlaps. For example, let ABCD be a small square polygon and consider its complement DCBA. These two polygons do not intersect and their union is the entire S2 sphere. Now consider the zig-zag polyline ABDC: its intersection with the polygon ABCD is the polyline ABCD (, whereas its intersection with the polygon DCBA is DC. Together these two polylines {ABD, DC} can be unioned to obtain the original polyline ABDC without gaps or overlaps.

All types of degeneracies are redundant in the semi-open model. In other words, degeneracies never affect point or edge containment but simply add to the boundary of the geometry. For example, a semi-open point polyline AA does not contain the point A (just like the open model), semi-open degenerate polygon shells do not contain any points or edges (just like the open model), and semi-open degenerate polygon holes do not exclude any points or edges (just like the closed model). All of these degeneracies simply add to the boundary of the geometry without affecting its interior or exterior. The semi-open model is purest of the three models in that shells and holes are treated symmetrically.

Boolean Operations

Many properties of boolean operations involving degeneracies follow naturally from the definitions above. Here we summarize the additional principles that define the results of such operations (intersection, union, difference, and symmetric difference).

Operations on geometry of a given dimension yields geometry of the same dimension. So for example, a polyline AB intersected with a polyline BC under the closed model yields the degenerate polyline BB rather than a point. Similarly a polygon ABC intersected with an abutting polygon CBD under the closed model yields the degenerate polygon shell BC, not a polyline. And symmetrically, a polygon hole ACB unioned with an abutting polygon hole BCD under the open model yields a degenerate polygon hole BC (which has no alternative representation). This behavior is exactly what allows S2 to avoid the serious problems of dimension reduction.

In all boundary models, subtracting an object from itself yields the empty set. This is true even for redundant degeneracies that contain no points or edges, such as subtracting a point polyline AA from itself in the open boundary model.

In all boundary models, intersecting or unioning an object with itself yields the original object(s). The reason for the plural is that points and polyline edges are treated as multisets. For example, if a point A is intersected with another point A, the result is two points {A, A}. (Note that such duplicates can easily be filtered away by choosing appropriate options in the S2Builder output layer.) Again, this is true even for redundant degeneracies that contain no points or edges, so for example a point shell A intersected with itself in the open boundary model yields the same point shell A, and a point polyline AA intesected with itself in the open boundary model yields two point polylines {AA, AA} (neither of which contains the point A).

Intersecting geometry of different dimensions yields only the lower-dimensional geometry. This is consistent with the principles described under Semantics of Degeneracies above. For example, intersecting a point A with a closed polygon ABC yields only the point A, not a point polygon shell A as well. Of course, if the input geometries do not intersect then the result is empty. For example intersecting a point A with an open polygon ABC yields nothing at all.

Unioning geometry of different dimensions yields only the higher-dimensional geometry. So for example, unioning a point A with a closed polyline AA yields only the polyline AA. Of course if the input geometries do not intersect then the result is simply the collection of both inputs. For example the union of a point A with an open or semi-open polyline AA is the collection of both objects {A, AA} since they do not intersect.

Unioning points with other points, or polylines with other polylines, yields a multiset. For example, the union of two identical points A is the multiset {A, A}, and the union of two identical polylines ABC is the multiset {ABC, ABC}. As noted earlier this behavior makes it possible to reconstruct time series such as GPS tracks accurately. Unwanted duplicates can be filtered by choosing appropriate options in the S2Builder output layer.

Semi-Open Semantics

The semi-open boundary model has a few properties that deserve further explanation.

In the semi-open model, complementary degeneracies do not intersect. So for example, a point shell A does not intersect a point hole A, and a sibling pair shell AB does not intersect a sibling pair hole AB.

In the semi-open model, a point polyline AA or point shell A intersects another different geometry if and only if that other geometry contains the point A. So for example, a point shell A intersects a semi-open polygon ABC if and only if ABC contains its vertex A under the usual semi-open rules. In other words, positive point degeneracies are treated exactly like points (except when a degeneracy is intersected with itself—see above).

The purpose of this rule is to ensure that the defining property of the semi-open model (i.e. the ability to cover a region without overlaps or gaps) applies to point degeneracies as well as points. For example, if a degenerate polyline AA or degenerate shell A is intersected with a set of semi-open polygons that tile the region around a shared vertex A, then exactly one of those polygons contains that degeneracy. Similarly, if a semi-open polyline ABC is expressed as the disjoint union of sub-polylines {AB, BC} then exactly one of those polylines contains the degenerate polyline BB (in particular, BC contains BB while AB does not).

As noted above, the only way that point degeneracies are handled differently than points is with respect to self-intersection. Specifically, in the semi-open model:

• A point A intersected with a polyline AA or point shell A is empty (since the latter objects do not contain any points).
• A polyline AA intersected with a point shell A is empty (since the point shell contains no points or polylines).
• And yet, a point A, polyline AA, or point shell A intersected with itself yields the original object.

These statements are not contradictory; there is in fact a valid representation of these objects as infinitesimal regions that yields this behavior.

The complementary property is that the union of a point hole A with another different geometry eliminates the hole if and only if that other geometry contains the point A. So for example, a point hole A unioned with the semi-open polygon ABC eliminates the hole from the result if and only if ABC contains its vertex A.

A similar property for edge degeneracies is that sibling pair shells do not intersect abutting polygons. So for example, a shell BC does not intersect the polygon ABC or CBD. The best way to think about this is that the intersection region is only one-half of the sibling pair degeneracy region and therefore cannot be represented. The complementary property is that the union of a sibling pair hole with an abutting polygon eliminates the hole. So for example, the union of the polygon ABC with the full sphere except for the hole BC is the full sphere.

Limitations

It is important to realize that in some cases the true mathematical result of an operation cannot be represented. This is true even when no degeneracies are present.

For example, consider a triangle ABC that contains a nested triangle DEF, and suppose that DEF is subtracted from ABC in the CLOSED boundary model. Ideally the result would contain the edges {AB, BC, CA} but not the edges {DE, EF, FD} (since the latter three edges belong to DEF, which has been subtracted). However it is not possible to represent this in the CLOSED model since all of these edges are on the boundary of the result.

Similarly, results involving degeneracies sometimes cannot be exactly represented. In such cases the preference is always to correctly represent the interior of the result (including infinitesimal interiors) even when the boundary of the result cannot be perfectly represented.

For example, consider subtracting a point shell D from the interior of a closed polygon ABC. The result is defined to be the polygon {ABC, D} where D is now a point hole. This may seem strange since (1) the result still contains the point D because the boundary is closed, and (2) there is no obvious reason to include D in the result because point holes exclude no points under the closed model (i.e., they are redundant). And yet this is the only correct result under the S2 degeneracy model, because the original point shell D is considered to include an infinitesimal two-dimensional area near the point D in addition to D itself. The only way to exclude this infinitesimal two-dimensional area from the result is to include D as a point hole.